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Convergence and Stability of Solutions obtained by the Method of Meshless Fundamental Solutions
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| Organization: | Positive Corporation Limited |
| Organization: | Wessex Institute of Technology |
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2002 Applications of Computer Algebra Conference (ACA '2002)
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Volos, Greece
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In this paper we use the method of external source collocation to solve a discretised boundary value problem , where U is the potential in a simply-connection region R, subject to a mixture of Neumann and Dirichlet boundary conditions. We aim to clarify previous discussions on how many sources are required to give sufficient accuracy and where they should be placed relative to the boundary. Such information has, hitherto, been anecdotal and somewhat subjective.
We present a general proof to show that a sequence of solutions to this boundary value problem converges as the sources are moved further from the boundary, provided that certain conditions on the placement of sources are satisfied, and that at least one of the boundary conditions is a Dirichlet type. In the case where only Neumann boundary conditions apply, convergence to a correct value is not assured, and we provide an explanation for this.
These theoretical results are backed up by numeric computations, for which Mathematica is invaluable, mainly because it allows numerical computation in severely ill-conditioned systems. Mathematica also provides a means of fast prototyping and allows a limited amount of symbolic computation to be done in simple cases. A specific example illustrates the difficulty of applying symbolic computation to this type of problem. If a limited number of sources are placed around a square boundary, Mathematica can derive symbolic expressions for the potential and flux at an interior point of the square, and can also derive the limit of these expressions as the sources move towards infinity. The computations are too slow if rational arithmetic is used, but a mixture of symbolic and floating-point computation can be successful. As the sources are moved further from the boundary, severe ill-conditioning limits the accuracy of other numerical programming languages (Fortran, C++ etc.), and necessitates the use of singular value decomposition to invert large matrices.
We conclude that there should be sufficient sources to accurately reflect the degrees of freedom of the problem (roughly one per boundary element), and that the further they are from the boundary, the more accurate the result.
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symbolic computation, numerical computation, prototyping, floating-point computation
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